From 5b08f435327695bb633cd21ae8252b25528de3f6 Mon Sep 17 00:00:00 2001
From: Connor Moore <connor@hhmoore.ca>
Date: Mon, 23 Mar 2026 01:40:30 -0400
Subject: New report and code for final submission.

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+\chapter{Introduction}
+\section{Basic Physics}
+This project presents a Langevin dynamics simulation of $N$ particles undergoing Brownian motion, confined in a two-dimensional square box of side length $L = 1.0$ with reflective boundary conditions. The primary objective is to study particle motion through the root-mean-square displacement (RMSD), which allows identification of ballistic ($\sim t^2$) and diffusive ($\sim t$) regimes. A secondary objective is computational: the most expensive component of the simulation is the evaluation of pairwise particle interactions arising from the WCA potential, which scales as $\mathcal{O}(N^2)$ per timestep. To address this, the program is structured using a domain decomposition algorithm to reduce computational cost and enable efficient parallelization with OpenMP. Brownian motion describes particles suspended in a fluid medium and subject to random thermal forces arising from microscopic collisions \cite{feynman41}. The equation of motion for a single particle is given by the Langevin equation:
+
+\begin{equation}
+  m\ddot{\mathbf{x}} = -\nabla U(\mathbf{x}) - \gamma \dot{\mathbf{x}} + 
+  \sqrt{2m\gamma k_BT}\, \mathbf{R}(t)
+\end{equation}
+
+where $-\nabla U(\mathbf{x})$ is the conservative WCA force, $-\gamma\dot{\mathbf{x}}$ represents viscous drag, and $\mathbf{R}(t)$ is a stochastic noise term whose strength scales with temperature. At short timescales, particle motion is ballistic, retaining memory of the initial velocity, while at longer timescales the motion becomes diffusive due to the randomizing effect of the noise. The simulation is implemented using the velocity-Verlet integration scheme, with random forces generated using the Box--Muller transform and initial velocities sampled from the Maxwell--Boltzmann distribution.
+
+Pairwise particle interactions are modelled using the Weeks--Chandler--Andersen (WCA) potential, the purely repulsive truncation of the Lennard-Jones potential. The force between particles $i$ and $j$ is given by
+\begin{equation}
+  \mathbf{F}_{ij} = -\nabla V(r_{ij}),
+\end{equation}
+
+where the Lennard-Jones potential is
+
+\begin{equation}
+  V(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - 
+  \left(\frac{\sigma}{r}\right)^6 \right].
+\end{equation}
+
+The interaction is truncated at the cutoff distance $r_c = 2^{1/6}\sigma$, corresponding to the minimum of the potential; for $r > r_c$ both the potential and force are set to zero.
+
+%\section{Boundary Conditions}
+%Various boundary conditions (BCs) exist when considering particle transport problems. The most common types include vacuum BCs, where particles are considered finished once they leave the domain, reflective BCs, where particles bounce off the domain boundary, and 
+%periodic BCs, where a particle leaving one side re-enters from the opposite side \cite{demaziere2020}. This simulation employs specular reflective BCs on all sides of the box, illustrated in Figure \ref{fig:reflective-BC}.
+
+%\begin{figure}[H]
+%    \centering
+%    \input{Figures/reflectiveBC.pdf_tex}
+%    \caption{Working principle of a specular reflective boundary condition. Red: A particle 
+%    penetrates the boundary after a discrete time step. Magenta: The particle position in 
+%    the $L$-direction is adjusted as if it had `bounced' off of the boundary during the 
+%    time step. Blue: The particle continues being modeled after the correction in the next 
+%    time step.}
+%    \label{fig:reflective-BC}
+%\end{figure}
\ No newline at end of file
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